NUMERICALSTUDY OFQUANTIZED VORTEX INTERACTION IN … · (also called as winding number or index)...

NUMERICAL STUDY OF QUANTIZED VORTEX INTERACTION IN

NONLINEAR SCHRODINGER EQUATION ON BOUNDED DOMAIN∗

WEIZHU BAO† AND QINGLIN TANG†

Abstract. In this paper, we study numerically quantized vortex dynamics and their interactionof the two-dimensional (2D) nonlinear Schrodinger equation (NLSE) with a dimensionless parameterε > 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. Webegin with a review of the reduced dynamical laws for time evolution of quantized vortex centers andshow how to solve these nonlinear ordinary differential equations numerically. Then we outline someefficient and accurate numerical methods for discretizing the NLSE on either a rectangle or a diskunder either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient andaccurate numerical methods for NLSE and the reduced dynamical laws, we simulate quantized vortexinteraction of NLSE with different ε and different initial setups including single vortex, vortex pair,vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduceddynamical laws, and examine the validity of the reduced dynamical laws. Finally, we investigateradiation and generation of sound waves as well as their impact on vortex interaction in the NLSEdynamics.

Key words. Nonlinear Schrodinger equation, quantized vortex, Dirichlet boundary condition,homogeneous Neumann boundary condition, reduced dynamical laws, time splitting.

AMS subject classifications. 35Q55, 65M06, 65M70, 81Q05, 82D50

1. Introduction. Vortices are those waves that possess phase singularities (topo-logical defect) and rotational flows around the singular points. They arise in manyphysical areas of different scale and nature ranging from liquid crystals and super-fluids to non-equilibrium patterns and cosmic strings [12, 30]. Quantized vortices intwo dimensions (2D) are those particle-like vortices, whose centers are zeros of theorder parameter, possessing localized phase singularity with the topological charge(also called as winding number or index) being quantized. They have been widelyobserved in many different physical systems, such as liquid helium, type-II supercon-ductors, atomic gases and nonlinear optics [1, 4, 11, 20, 25]. Quantized vortices arekey signatures of superconductivity and superfluidity and their study retains as oneof the most important and fundamental problems since they were predicted by LarsOnsager in 1947 in connection with superfluid Helium.

In this paper, we consider and study numerically vortex dynamics and interactionsin the 2D nonlinear Schrodinger equation (NLSE) called also as the Gross-Pitaevskiiequation (GPE), which is a fundamental equation for modeling and understandingsuperfluids [2, 14, 15, 17, 31]:

(1.1) i∂tψε(x, t) = ∆ψε +

1

ε2(1− |ψε|2)ψε, x ∈ Ω, t > 0,

with initial condition

(1.2) ψε(x, 0) = ψε0(x), x ∈ Ω,

and either Dirichlet boundary condition (BC)

(1.3) ψε(x, t) = g(x) = eiω(x), x ∈ ∂Ω, t ≥ 0,

∗This research was supported by the Singapore A*STAR SERC “Complex Systems” ResearchProgramme grant R-146-000-171-305.

†Department of Mathematics and Center for Computational Science and Engineering, Na-tional University of Singapore, Singapore 119076 (URL: http://www.math.nus.edu.sg/˜bao/,[email protected] (W. Bao); [email protected] (Q. Tang)).

1

2 W. Bao and Q. Tang

or homogeneous Neumann BC

(1.4)∂ψε(x, t)

∂n= 0, x ∈ ∂Ω, t ≥ 0.

Here, Ω ⊂ R2 is a 2D simple connected and bounded domain, t is time, x = (x, y) ∈ R

2

is the Cartesian coordinate vector, ψε := ψε(x, t) is a complex-valued function de-scribing the ‘order parameter’ for a superfluid, ω is a given real-valued function, ψε

0

and g are given smooth and complex-valued functions satisfying the compatibilitycondition ψε

0(x) = g(x) for x ∈ ∂Ω, n = (n1, n2) and n⊥ = (−n2, n1) ∈ R2 satis-

fying |n| =√

n21 + n2

2 = 1 are the outward normal and tangent vectors along ∂Ω,respectively, ε > 0 is a given dimensionless constant.

Denote the Gross-Pitaevskii (GP) or Ginzburg-Landau (GL) functional (‘energy’)as [9, 16, 21]

(1.5) Eε(t) :=

∫

Ω

[

|∇ψε(x, t)|2 +1

2ε2(

|ψε(x, t)|2 − 1)2]

dx = Eεkin(t) + Eε

int(t),

then it is easy to see that, for the NLSE (1.1) with either Dirichlet BC (1.3) orhomogeneous Neumann BC (1.4) for general domain Ω, or periodic BC when Ω is arectangle, the GP functional Eε(t) is conserved, i.e. Eε(t) ≡ Eε(0) for t ≥ 0.

There have been several analytical and numerical studies recently that deal withquantized vortex states of the NLSE (1.1) and their interaction in the whole spaceR

2 or on bounded domains under different scalings regarding to the distances be-tween different vortices. Based on a formal analysis, Fetter [13] predicted that, tothe leading order, the dynamics of vortices in the NLSE (1.1) would be governed bythe same law as that in the ideal incompressible fluid. Then, the same prediction wasgiven mathematically by Neu [25] using the method of matched asymptotics. In Neu’swork [25], he found vortex states of the NLSE (1.1) in the whole space R2 with ε = 1for superfluidity and conjectured the stability of these states under NLSE dynamicsas an open problem [25]. Based on his conjecture on the stability, he obtained for-mally the reduced dynamical laws governing the motion of the vortex centers underthe assumption that these vortices are distinct and well-separated, i.e. the reduceddynamical laws are asymptotically valid when the distances between vortex centersbecome larger and larger [25]. In fact, under this scaling, the vortex core size is O(1).Based on the reduced dynamical laws which is a set of ordinary differential equations(ODEs) for the vortex centers, one can obtain that two vortices with opposite windingnumbers (vortex dipole or vortex-antivortex) move parallel, while they rotate along acircle if they have the same winding number (vortex pair). However, these ODEs areonly correct up to the leading order. Corrections to this leading order approximationdue to radiation and/or related questions when long-time dynamics of vortices is con-sidered still remain as important open problems. In fact, using the method of effectiveaction and geometric solvability, Ovichinnikov and Sigal confirmed Neu’s approxima-tion and derived some leading radiative corrections [28, 29] based on the assumptionthat the vortices are well separated, which was extended by Lange and Schroers [19]to study the dynamics of overlapping vortices. Recently, by proposing efficient andaccurate numerical methods for discretizing the NLSE in the whole space, Zhang etal. [35, 36] compared the dynamics of quantized vortices from the reduced dynam-ical laws obtained by Neu with those from the direct numerical simulation resultsfrom NLSE under different parameter and/or initial setups. They solved numericallyNeu’s open problem on the stability of vortex states under the NLSE dynamics, i.e.

QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAIN 3

vortices with winding number m = ±1 are dynamically stable, and resp., |m| > 1dynamically unstable [35, 36], which agree with those conclusions derived asymptoti-cally by Ovchinnikov and Sigal [27]. In addition, they identified numerically differentparameter regimes where the reduced dynamical laws agree qualitatively and/or quan-titatively as well as fail to agree with those from the NLSE dynamics on quantizedvortices and their interaction [35, 36].

Inspired by Neu’s work, many papers have been dedicated to the study of the vor-tex states and dynamics governed by the NLSE (1.1) on a bounded domain Ω ⊂ R

2

with introducing a small dimensionless parameter 0 < ε < 1 which is proportional tothe core size of a vortex. Under this scaling, the core size of each vortex is O(ε) and thedistances between vortex centers initially are O(1). Mironescu [24] investigated sta-bility of the vortices in (1.1) with (1.3) and showed that for fixed winding number m:a vortex with |m| = 1 is always dynamical stable; while for those of winding number|m| > 1, there exists a critical εcm such that if ε > εcm, the vortex is stable, otherwiseunstable. Mironescu’s results were then improved by Lin [23] using the spectrum of alinearized operator. Subsequently, Lin and Xin [21] studied the vortex dynamics ona bounded domain with either Dirichlet or Neumann BC, which was further investi-gated by Jerrard and Spirn [16] using different methods. In addition, Colliander andJerrard [9,10] studied the vortex structures and dynamics on a torus or under periodicBC. In these studies, the authors derived the reduced dynamical laws which governthe dynamics of vortex centers under the NLSE dynamics when ε → 0 with fixeddistances between different vortex centers initially. They obtained that to the leadingorder the vortices move according to the Kirchhoff law in the bounded domain case.However, these reduced dynamical laws cannot indicate radiation and/or sound prop-agations created by highly co-rotating or overlapping vortices. It remains as a veryfascinating and fundamental problem to understand the vortex-sound interaction [26],and how the sound waves modify the motion of vortices [14].

Very recently, the authors designed some efficient and accurate numerical methodsfor studying vortex dynamics and interactions in the Ginzburg-Landau equation onbounded domains with either Dirichlet or Neumann BCs [5]. These numerical methodscan be extended and applied for studying the rich and complicated phenomena relatedto vortex dynamics in the NLSE (1.1) with either Dirichlet BC (1.3) or homogeneousNeumann BC (1.4) on bounded domains. In fact, the main aim of this paper is:(i) to present efficient and accurate numerical methods for discretizing the reduceddynamical laws and the NLSE (1.1) on bounded domains under different BCs, (ii) tounderstand numerically how the boundary condition and radiation as well as geometryof the domain affect vortex dynamics and interction, (iii) to study numerically vortexinteraction in the NLSE dynamics and/or compare them with those from the reduceddynamical laws with different initial setups and parameter regimes, and (iv) to identifycases where the reduced dynamical laws agree qualitatively and/or quantitatively aswell as fail to agree with those from NLSE on vortex interaction.

The rest of the paper is organized as follows. In section 2, we briefly reviewthe reduced dynamical laws of vortex interaction under the NLSE (1.1) with eitherDirithlet or homogeneous Neumann BC and present numerical methods to discretizethem. In section 3, efficient and accurate numerical methods are briefly outlined fordiscretizing the NLSE on bounded domains with different BCs. In section 4, numericalresults are reported for vortex interaction of NLSE under Dirichlet BC, and similarresults for NLSE under homogeneous Neumann BC are reported in section 5. Finally,some conclusions are drawn in section 6.


2. The reduced dynamical laws and their discretization. In this section,we review two different forms of the reduced dynamical laws for the dynamics ofvortex centers in the NLSE (1.1) with either Dirichlet or homogeneous Neumann BC,show their equivalence and present efficient numerical methods to discretize them.

We assume that, in the initial data ψε0, there are exactly M isolated and distinct

vortices whose centers are located at x01 = (x01, y

01), x

02 = (x02, y

02), . . ., x

0M = (x0M , y

0M )

with winding numbers n1, n2, . . ., nM, respectively. The winding number of each

vortex can be chosen as either 1 or −1, i.e. nj = 1 or −1 for j = 1, 2, . . . ,M . At timet ≥ 0, the M isolated and distinct vortex centers are located at x1(t) = (x1(t), y1(t)),x2(t) = (x2(t), y2(t)), . . ., and xM (t) = (xM (t), yM (t)). Denote

X0 := (x01,x

02, . . . ,x

0M ), X := X(t) = (x1(t),x2(t), . . . ,xM (t)), t ≥ 0,

then the renormalized energy associated to the M vortex centers is defined as [7, 20]

(2.1) W (X) :=W (X(t)) = −∑

1≤j 6=l≤M

njnl ln |xj(t)− xl(t)| , t ≥ 0.

2.1. Under the Dirichlet BC. For the NLSE (1.1) with Dirichlet BC (1.3),when ε → 0, two different forms of the reduced dynamical laws have been obtainedin the literatures for governing the motion of the M vortex centers.

The first one which is widely used has been derived formally and rigorously inthe literatures, see for instance [7, 9, 18, 22] and references therein:

(2.2) xj(t) = −njJ∇xj[W (X) +Wdbc(X)] , j = 1, . . . ,M, t > 0,

with initial condition

(2.3) xj(0) = x0j , j = 1, 2, . . . ,M.

Here, J is a 2× 2 symplectic matrix defined as

J =

(

0 1−1 0

)

,

and the renormalized energy Wdbc(X) comes from the effect of the Dirichlet BCassociated to the M vortex centers X = X(t) and it is defined as [7, 20]:

Wdbc(X) = −

M∑

j=1

njR(xj ;X) +

∫

∂Ω

R(x;X) +

M∑

j=1

nj ln |x− xj |

∂n⊥ω(x)

2πds,

where, for any fixed X ∈ ΩM , R(x;X) is a harmonic function in x, i.e.

(2.4) ∆R(x;X) = 0, x ∈ Ω,

with the following Neumann BC

(2.5)∂R(x;X)

∂n= ∂n⊥

ω(x)−∂

∂n

M∑

l=1

nl ln |x− xl|, x ∈ ∂Ω.

Using an identity in [7] (see Eq. (51) on page 84),

∇xj[W (X) +Wdbc(X)] = −2nj∇x

R(x;X) +M∑

l=1&l 6=j

nl ln |x− xl|

x=xj

,


the reduced dynamical law (2.2) can be simplified, for 1 ≤ j ≤M , as

xj(t) = 2J

∇xR (x;X) |x=xj(t) +

M∑

l=1&l 6=j

nl

xj(t)− xl(t)

|xj(t)− xl(t)|2

, t > 0.(2.6)

The second one was obtained by Lin and Xin [21], for 1 ≤ j ≤M , as

xj(t) = 2J

J∇xH (x;X) |x=xj(t) +

M∑

l=1&l 6=j

nl

xj(t)− xl(t)

|xj(t)− xl(t)|2

, t > 0;(2.7)

where for any fixed X ∈ ΩM , H(x;X) is a harmonic function in x and satisfies thefollowing BC

(2.8)∂H(x;X)

∂n⊥= ∂n⊥

ω(x)−∂

∂n

M∑

l=1

nl ln |x− xl|, x ∈ ∂Ω.

In the above two different forms of the reduced dynamical laws for NLSE, althoughthe two harmonic functions R(x;X) and H(x;X) satisfy different BCs, in fact, theyare equivalent. Actually, they are both equivalent to the following reduced dynamicallaw: for 1 ≤ j ≤M ,

xj(t) = 2J

J∇xQ (x;X) |x=xj(t) +

M∑

l=1&l 6=j

nl

xj(t)− xl(t)

|xj(t)− xl(t)|2

, t > 0;(2.9)

where for any fixed X ∈ ΩM , Q(x;X) is a harmonic function in x and satisfies thefollowing Dirichlet BC

(2.10) Q(x;X) = ω(x)−

M∑

l=1

nlθ(x − xl), x ∈ ∂Ω,

with the function θ : R2 → [0, 2π) being defined as

(2.11) cos(θ(x)) =x

|x|, sin(θ(x)) =

y

|x|, 0 6= x = (x, y) ∈ R

2.

Lemma 2.1. For any fixed X ∈ ΩM , we have the following identity

(2.12) J∇xQ (x;X) = ∇xR (x;X) = J∇xH (x;X) , x ∈ Ω,

which immediately implies the equivalence of the three reduced dynamical laws (2.6),(2.7) and (2.9).

Proof. For any fixed X ∈ ΩM , since Q is a harmonic function, there exists afunction ϕ1(x) such that

J∇xQ (x;X) = ∇ϕ1(x), x ∈ Ω.

Thus, ϕ1(x) satisfies the Laplace equation

(2.13) ∆ϕ1(x) = ∇ · (J∇xQ(x;X)) = ∂yxϕ1(x)− ∂xyϕ1(x) = 0, x ∈ Ω,


with the following Neumann BC

(2.14) ∂nϕ1(x) = (J∇xQ(x;X)) · n = ∇xQ(x;X) · n⊥ = ∂n⊥Q(x;X), x ∈ ∂Ω.

Noticing (2.10), we obtain for x ∈ ∂Ω

∂nϕ1(x) = ∂n⊥ω(x)−

∂

∂n⊥

M∑

l=1

nlθ(x− xl) = ∂n⊥ω(x) −

∂

∂n

M∑

l=1

nl ln |x− xl|.

Combining the above equality with (2.13), (2.4) and (2.5), we get

∆(R(x;X)− ϕ1(x)) = 0, x ∈ Ω; ∂n (R(x;X)− ϕ1(x)) = 0, x ∈ ∂Ω.

Thus

R(x;X) = ϕ1(x) + constant, x ∈ Ω,

which immediately implies the first equality in (2.12). The second equality in (2.12)can be proved in a similar way and we omit it here for brevity.

2.2. Under the homogeneous Neumann BC. Similarly, for the NLSE (1.1)with the homogeneous Neumann BC (1.4), when ε → 0, there are also two differentforms of the reduced dynamical laws that govern the motion of the M vortex centers.

Again, by introducing the renormalized energy Wnbc which comes from the effectof the the homogeneous Neumann BC associated to the M vortex centers X = X(t)

(2.15) Wnbc(X) = −M∑

j=1

njR(xj ;X),

where, for any fixed X ∈ ΩM , R(x;X) is a harmonic function in x and satisfies thefollowing Dirichlet BC

(2.16) R(x;X) = −

M∑

l=1

nl ln |x− xl|, x ∈ ∂Ω,

the first one has been derived formally and rigourously by several authors in theliteratures [9, 16, 18] as

(2.17) xj(t) = −njJ∇xj[W (X) +Wnbc(X)] , j = 1, . . . ,M, t > 0.

Using the following identity

(2.18) ∇xj[W (X) +Wnbc(X)] = −2nj∇x

R(x;X) +

M∑

l=1&l 6=j

nl ln |x− xl|

xj

,

the above reduced dynamical laws collapse, for 1 ≤ j ≤M , as

xj(t) = 2J

∇xR (x;X) |x=xj(t) +

M∑

l=1&l 6=j

nl

xj(t)− xl(t)

|xj(t)− xl(t)|2

, t > 0.(2.19)


Similarly, the second one was obtained by Lin and Xin [21], for 1 ≤ j ≤M , as

xj(t) = 2J

J∇xQ (x;X) |x=xj(t) +

M∑

l=1&l 6=j

nl

xj(t)− xl(t)

|xj(t)− xl(t)|2

, t > 0;(2.20)

where, for any fixed X ∈ ΩM , Q(x;X) is a harmonic function in x and satisfies thefollowing Neumann BC

(2.21)∂Q(x;X)

∂n= −

∂

∂n

M∑

l=1

nlθ(x− xl), x ∈ ∂Ω.

Similar to the proof of Lemma 2.1, we can establish the equivalence of the abovetwo different forms of the reduced dynamical laws.

Lemma 2.2. The reduced dynamical laws (2.19) and (2.20) are equivalent.In order to compare the solution of the reduced dynamical laws (2.6) or (2.9) and

(2.19) or (2.20) with those from the NLSE under Dirichlet or homogeneous NeumannBC, respectively, the ODEs (2.6) or (2.9) and (2.19) or (2.20) are discretized by thestandard fourth order Runge-Kutta method. For each fixed X ∈ ΩM , when thedomain Ω is a rectangle, the Laplace equation (2.4) with BCs (2.5) or (2.10) or (2.16)or (2.21) is discretized by the standard second order finite difference method; andrespectively, when the domain Ω is a disk, they are discretized in θ-direction viathe Fourier pseudospectral method and in r-direction via the finite element method(FEM) with (r, θ) the polar coordinates. For details, we refer to [5] and omit themhere for brevity.

3. Numerical methods. In this section, we outline briefly some efficient andaccurate numerical methods for discretizing the NLSE (1.1) in either a rectangle or adisk with initial condition (1.2) and either Dirichlet or homogeneous Neumann BC.The key ideas in our numerical methods are based on: (i) applying a time-splittingtechnique which has been widely used for nonlinear partial differential equations [34] todecouple the nonlinearity in the NLSE [4,6,35]; and (ii) adapt proper finite differenceand/or finite element method to discretize a free Schrodinger equation [4, 5].

Let τ := t > 0 be the time step size, denote tn = nτ for n ≥ 0. For n = 0, 1, . . .,from time t = tn to t = tn+1, the NLSE (1.1) is solved in two splitting steps. Onefirst solves

(3.1) i∂tψε(x, t) =

1

ε2(1− |ψε|2)ψε, x ∈ Ω, t ≥ tn,

for the time step of length τ , followed by solving

(3.2) i∂tψε(x, t) = ∆ψε, x ∈ Ω, t ≥ tn,

for the same time step. The discretization of (3.2) will be outlined later. For t ∈[tn, tn+1], we can easily obtain the following ODE for ρ(x, t) = |ψε(x, t)|2:

(3.3) ∂tρ(x, t) = 0, t ≥ tn, x ∈ Ω,

which implies that

(3.4) ρ(x, t) = ρ(x, tn), t ≥ tn, x ∈ Ω.


Plugging (3.4) into (3.1), we can integrate it exactly to get

(3.5) ψε(x, t) = ψε(x, tn) exp

[

−1

ε2(1 − |ψε(x, tn)|

2)(t− tn)

]

, t ≥ tn, x ∈ Ω.

We remark here that, in practice, we always use the second-order Strang splitting[34], that is, from time t = tn to t = tn+1: (i) evolve (3.1) for half time step τ/2 withinitial data given at t = tn; (ii) evolve (3.2) for one step τ starting with the new data;and (iii) evolve (3.1) for half time step τ/2 again with the newer data.

When Ω = [a, b]× [c, d] is a rectangular domain, we denote hx=b−aN

and hy=d−cL

with N and L being two even positive integers as the mesh sizes in x−direction andy−direction, respectively. Similar to the discretization of the gradient flow with con-stant coefficient [5], when the Dirichlet BC (1.3) is used for the equation (3.2), itcan be discretized by using the 4th-order compact finite difference discretization forspatial derivatives followed by a Crank-Nicolson (CNFD) scheme for temporal deriva-tive [5]; and when homogeneous Neumann BC (1.4) is used for the equation (3.2), itcan be discretized by using cosine spectral discretization for spatial derivatives fol-lowed by integrating in time exactly [5]. The details are omitted here for brevity.Combining the CNFD and cosine psedudospectral discretization for Dirichlet and ho-mogeneous Neumann BC, respectively, with the second order Strang splitting, we canobtain time-splitting Crank-Nicolson finite difference (TSCNFD) and time-splittingcosine psedudospectral (TSCP) discretizations for the NLSE (1.1) on a rectangle withDirichlet BC (1.3) and homogeneous Neumann BC (1.4), respectively. Both TSCNFDand TSCP discretizations are unconditionally stable, second order in time, the mem-ory cost is O(NL) and the computational cost per time step is O (NL ln(NL)). Inaddition, TSCNFD is fourth order in space and TSCP is spectral order in space.

When Ω = x | |x| < R := BR(0) is a disk with R > 0 a fixed constant.Similar to the discretization of the GPE with an angular momentum rotation [3, 4,35] and/or the gradient flow with constant coefficient [5], it is natural to adopt thepolar coordinate (r, θ) in the numerical discretization by using the standard Fourierpseduospectral method in θ-direction [33], finite element method in r-direction, andCrank-Nicolson method in time [3, 4, 35]. Again, the details are omitted here forbrevity.

4. Numerical results under Dirichlet BC. In this section, we report numer-ical results for vortex interactions of the NLSE (1.1) under the Dirichlet BC (1.3)and compare them with those obtained from the reduced dynamical laws (2.6) with(2.3). We study how the dimensionless parameter ε, initial setup, boundary value andgeometry of the domain Ω affect the dynamics and interaction of vortices. For a givenbounded domain Ω, the NLSE (1.1) is unchanged by the re-scaling x → dx, t → d2tand ε → dε with d the diameter of Ω. Thus without lose of generality, hereafter,without specification, we always assume that the diameter of Ω is O(1). The functiong in the Dirichlet BC (1.3) is given as

g(x) = ei(h(x)+∑M

j=1njθ(x−x

0

j)), x ∈ ∂Ω,

and the initial condition ψε0 in (1.2) is chosen as

(4.1) ψε0(x) = ψε

0(x, y) = eih(x)M∏

j=1

φεnj(x − x0

j), x = (x, y) ∈ Ω,


−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

0 0.5 10

0.17

0.34

t

d 1ε

ε=1/32ε=1/50ε=1/80

0 0.8 1.60

0.06

0.12

t

d 1ε

0 0.5 10

0.2

0.4

t

d 1ε

ε=1/32ε=1/50ε=1/80

ε=1/16ε=1/40ε=1/80

Fig. 4.1: Trajectory of the vortex center in NLSE under Dirichlet BC when ε = 140

for Cases I-VI (from left to right and then from top to bottom in top two rows), anddε1 for different ε for Cases I,V&VI (from left to right in bottom row) in section 4.1.

where M > 0 is the total number of vortices in the initial data, the phase shift h(x)is a harmonic function, θ(x) is defined in (2.11) and for j = 1, 2, . . . ,M , nj = 1 or−1, and x0

j = (x0j , y0j ) ∈ Ω are the winding number and initial location of the j-th

vortex, respectively. Moreover, for j = 1, . . . ,M , the function φεnj(x) is chosen as a

single vortex centered at the origin with winding number nj = 1 or −1 which wascomputed numerically and depicted in section 4 in [5]. In addition, in the followingsections, we mainly consider six different modes of the phase shift h(x) :

• Mode 0: h(x) = 0, Mode 1: h(x) = x+ y,• Mode 2: h(x) = x− y, Mode 3: h(x) = x2 − y2,• Mode 4: h(x) = x2 − y2 + 2xy, Mode 5: h(x) = x2 − y2 − 2xy.

To simplify our presentation, for j = 1, 2, . . . ,M , hereafter we let xεj(t) and xr

j(t)be the j-th vortex center in the NLSE dynamics and the corresponding reduced dy-namical laws, respectively, and denote dεj(t) = |xε

j(t)− xrj(t)| as their difference. Fur-

thermore, in the presentation of figures, the initial location of a vortex with windingnumber +1, −1 and the location that two vortices merge are marked as ‘+’, ‘’ and‘⋄’, respectively. Finally, in our computations, if not specified, we take Ω = [−1, 1]2 in(1.1), mesh sizes hx = hy = ε

10 and time step τ = 10−6. The NLSE (1.1) with (1.3),(1.2) and (4.1) is solved by the method TSCNFD presented in section 3.

4.1. Single vortex. Here we present numerical results of the motion of a singlequantized vortex in the NLSE (1.1) dynamics and the corresponding reduced dynam-ics, i.e. we take M = 1, n1 = 1 in (4.1). To study how the initial phase shift h(x) and


−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

Fig. 4.2: Trajectory of the vortex center in NLSE dynamics under Dirichlet BC whenε = 1

64 (blue solid line) and from the reduced dynamical laws (red dash line) for CasesVI-XI (from left to right and then from top to bottom) in section 4.1.

−1 0 1−0.65

0

0.65

x

y

−1 0 1−1

0

1

x

y

−1 0 1−0.65

0

0.65

x

y

−1 0 1−1

0

1

x

y

Fig. 4.3: Trajectory of the vortex center in NLSE under Dirichlet BC when ε = 140 for

cases I, XII-XIII, VI and XIV-XV (from left to right and then from top to bottom)in section 4.1.

initial location of the vortex x01 affect the motion of the vortex and to understand the

validity of the reduced dynamical law, we consider the following 11 cases:• Case I-III: x0

1 = (0, 0), and h(x) is chosen as Mode 1, 2 and 3, respectively;• Case IV-VIII: x0

1 = (0.1, 0), and h(x) is chosen as Mode 1, 2, 3, 4 and 5,respectively;

• Case IX-XI: x01 = (0.1, 0.1), and h(x) is chosen as Mode 3, 4 and 5, respec-

tively.Moreover, to study the effect of domain geometry, we consider Ω of three types: typeI. a square Ω = [−1, 1]× [−1, 1], type II. a rectangle Ω = [−1, 1]× [−0.65, 0.65], and


type III. a unit disk Ω = B1(0). Thus we also study the following 4 additional cases:• Case XII-XIII: x0

1 = (0, 0), h(x) = x + y, Ω is chosen as type II and III,respectively;

• Case XIV-XV: x01 = (0.1, 0), h(x) = x2 − y2, Ω is chosen as type II and III,

respectively.Fig. 4.1 depicts trajectory of the vortex center when ε = 1

40 in (1.1) for CasesI-VI and dε1 with different ε for Cases I, V and VI. Fig. 4.2 shows trajectory of thevortex center when ε = 1

64 in (1.1) for cases VI-XI, while Fig. 4.3 shows that forCases XII-XVII when ε = 1

32 in (1.1). From Figs. 4.1-4.3 and additional numericalexperiments not shown here for brevity, we can draw the following conclusions:

(i). When h(x) ≡ 0, the vortex center doesn’t move and this is similar to the casein the whole space.

(ii). When h(x) = (x + by)(x − yb) with b 6= 0, the vortex does not move if

x01 = (0, 0), while it does move if x0

1 6= (0, 0) (cf. Fig. 4.1 Cases III and VI for b = 1).(iii). When h(x) 6= 0 and h(x) 6= (x + by)(x − y

b) with b 6= 0, in general, the

vortex center does move. For the NLSE dynamics, there exists a critical value εcdepending on h(x), x0

1 and Ω such that if ε < εc, the vortex will move periodically ina close loop (cf. Fig. 4.1), otherwise their trajectory will not be a close loop. Thisdiffers from the situation in the reduced dynamics significantly, where the trajectoryis always periodic (cf. Fig. 4.2 red dash line). Thus the reduced dynamical laws failqualitatively when ε > εc. It should be an interesting problem to find how this criticalvalue depends on h(x), x0

1 and the geometry of Ω.(iv). In general, the initial location, the geometry of the domain and the boundary

value will all affect the motion of the vortex (cf. Fig. 4.3).(v). When ε → 0, the dynamics of the vortex center in the NLSE dynamics

converges uniformly in time to that in the reduced dynamics (cf. Fig. 4.1 bottomrow) which verifies numerically the validation of the reduced dynamical laws.

4.2. Vortex pair. Here we present numerical results of the interaction of vortexpair under the NLSE (1.1) dynamics and its corresponding reduced dynamical laws,i.e. we take M = 2, n1 = n2 = 1, x0

2 = −x01 = (d0, 0) with 0 < d0 < 1 (i.e. the two

vortices are initially located symmetric on the x-axis), h(x) ≡ 0 in (4.1) and ε = 140

in (1.1). Fig. 4.4 depicts the trajectory of the vortex pair and time evolution of Eε(t),Eεkin(t), x

ε1(t), x

ε2(t) and d

ε1(t) for different d0.

From Fig. 4.4 and additional numerical results not shown here for brevity, wecan draw the following conclusions for the interaction of vortex pair under the NLSEdynamics (1.1) with Dirichlet BC:

(i). The total energy is conserved numerically very well during the dynamics.(ii). The initial locations of the two vortices affect significantly the motion of the

vortex pair. If the vortex pair is symmetric initially, then they move periodically andtheir trajectories are symmetric i.e. xε

1(t) = −xε2(t) for t ≥ 0. Furthermore, for both

the reduced dynamical law and NLSE dynamics, there exist critical values, say drcand dεc, respectively, such that if d0 < drc (or d0 < dεc in the NLSE dynamics), the twovortices will rotate with each other and move along a circle-like trajectory, otherwise,they will move along a crescent-like trajectory (cf. Fig. 4.4). We find numerically thecritical value drc ≈ 0.4923 and dεc for 0 < ε < 1 which are depicted in Fig. 4.5. Fromthese values, we can fit the following relation for drc and dεc:

dεc = drc + 2.11ε2.08, 0 < ε < 1.

(iii). For any fixed d0, the dynamics of the two vortex centers under the NLSE


−0.8 0 0.8−0.8

0

0.8

x

y

0 0.075 0.15−0.11

0.11

x 1ε or

x 2ε

t

0 0.075 0.15−0.11

0.11

t

y 1ε or

y 2ε

0 0.075 0.1531.6

34.4

37.2

t

E ε

k i n

E ε

0 0.075 0.150

0.1

0.2

t

d 1ε

ε=1/40ε=1/50ε=1/64

−1 0 1−1

0

1

x

y

0 2 4−1

1

t

x 1ε or

x 2ε

0 2 4−1

1

t

y 1ε or

y 2ε

0 2 423

25

27

t

E ε

k i n

E ε

0 2 40

0.47

0.94

t

d 1ε

ε=1/16ε=1/25ε=1/40

Fig. 4.4: Trajectory of the vortex pair (left column), time evolution of xε1(t) and xε

2(t)(second column), Eε(t) and Eε

kin(t) (third column) and dε1(t) (last column) for d0 = 0.1(top row) and d0 = 0.5 (bottom row) in section 4.2.

0.02 0.06 0.10

0.01

0.02

ε

d cε −dcr

−3.8 −2.8 −1.8−7

−5.5

−3.8

ln(ε)

ln(d

cε −dcr )

Fig. 4.5: Critical value dεc for the interaction of vortex pair of the NLSE (1.1) underDirichlet BC with h(x) = 0 in (4.1) for different ε in section 4.2.

dynamics converges uniformly in time to that of the reduced dynamical laws (cf. Fig.4.4) when ε → 0. However, for fixed ε, the reduced dynamical law fails qualitativelyto describe the motion of vortices if drc < d0 < dεc.

4.3. Vortex dipole. Here we present numerical results of the interaction of vor-tex dipole under the NLSE (1.1) dynamics and its corresponding reduced dynamicallaws, i.e. we take M = 2, h(x) ≡ 0, n1 = −n2 = −1, x0

2 = −x01 = (d0, 0) in (4.1) with

d0 = 0.5 and ε = 125 in (1.1). Fig. 4.6 depicts contour plots of |ψε(x, t)| at different

times, trajectory of the vortex dipole, time evolution of xε1(t), x

ε2(t) and d

ε1(t). From

Fig. 4.6 and additional numerical results not shown here for brevity, we can draw thefollowing conclusions:

(i). The total energy is conserved numerically very well during the dynamics.

(ii). The vortex dipole moves upward symmetrically with respect to y-axis andfinally merges and annihilates somewhere near the top boundary simultaneously. Thedistance between the merging place and the boundary is of O(ε) when ε is small.After merging, new waves will be created and reflected by the top boundary. Thenew waves will then move back into the domain and be reflected back into the domain


−1 0 1−1

0

1

x

y

0 0.15 0.30

0.04

0.08

t

d 1ε

ε=1/8

ε=1/16

ε=1/32

0 0.165 0.33−0.6

0.6

t

x 1ε or

x 2ε

0 0.165 0.330

1

t

y 1ε or

y 2ε

Fig. 4.6: Contour plots of |ψε(x, t)| at different times (top two rows), trajectory(bottom left), time evolution of xε

1(t) and xε2(t) (bottom middle) and dε1(t) (bottom

right) for the dynamics of a vortex dipole in section 4.3.

again when they hit the boundaries (cf. Fig. 4.6). Moreover, the vortex dipole inthe NLSE dynamics will always merge in some place near the top boundary for alld0. However, in the reduced dynamics, they never merge inside Ω, in fact, they willmove outside the domain before they merge. Hence, the reduced dynamical law failsquantitatively when the vortex dipole is near the boundary.

(iii). When ε → 0, the dynamics of the two vortex centers under the NLSEdynamics converges uniformly to that of the reduced dynamical laws (cf. Fig. 4.6)before they merge each other or near the boundary which verifies numerically thevalidation of the reduced dynamical laws in this case. In fact, based on our extensivenumerical experiments, the motion of the two vortex centers from the reduced dy-namical laws agree with those from the NLSE dynamics qualitatively when 0 < ε < 1and quantitatively when 0 < ε≪ 1 when they are not too close to the boundary.

4.4. Vortex lattice. Here we present numerical studies on the dynamics ofvortex lattices in the NLSE (1.1) with Dirichlet BC (1.3), i.e. choose the initial data(1.2) as (4.1) and study four cases:

Case I. M = 3, n1 = n2 = n3 = 1, x01 = −x0

3 = (d0, 0) and x02 = (0, 0).

Case II. M = 3, n1 = −n2 = n3 = 1, x01 = −x0

3 = (d0, 0) and x02 = (0, 0).

Case III. M = 4, n1 = n2 = −n3 = −n4 = 1, x01 = −x0

2 = (d1, 0) and x03 =

−x04 = (0, d2) with 0 < d1, d2 < 1.Case IV. Ω = B5(0), M = 9, n1 = n2 = · · · = n9 = 1 and the 9 vortex centers are

initially located on a 3× 3 uniform mesh points for the rectangle [−d0, d0]× [−d0, d0]


−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−0.4 0 0.4−0.4

0

0.4

x

y

0 0.15 0.31 0.456−0.3

0

0.3

0.6

0.9

t

x

1ε(t)

y1ε(t)

x2ε(t)

y2ε(t)

0.456 0.49 0.53 0.56−0.3

0

0.3

0.6

0.9

t

x

1ε(t)

y1ε(t)

x2ε(t)

y2ε(t)

0.56 0.89 1.21 1.54−0.3

0

0.3

0.6

0.9

t

x

1ε(t)

y1ε(t)

x2ε(t)

y2ε(t)

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

0 0.37 0.73 1.1−0.3

0

0.3

0.6

0.9

t

x3ε(t)

y3ε(t)

x2ε(t)

y2ε(t)

1.1 1.15 1.2 1.25−0.5

0

0.5

1

t

x

3ε(t)

y3ε(t)

x2ε(t)

y2ε(t)

1.25 1.83 2.42 3−0.2

0

0.2

0.4

t

x

1ε(t)

y1ε(t)

Fig. 4.7: Trajectory of the vortex xε1 (blue line), xε

2 (dark dash-dot line) and xε3 (red

dash line) (first and third rows) and their corresponding time evolution (second andfourth rows) for Case I (top two rows) and Case II (bottom two rows) for small time(left column), intermediate time (middle column) and large time (right column) withε = 1

40 and d0 = 0.25 in section 4.4.

with 0 < d0 < 1.

Fig. 4.7 depicts the trajectory and time evolution of x01(t), x

02(t) and x0

3(t) in theNLSE dynamics for Cases I and II. Fig. 4.8 shows contour plots of |ψε| at differenttimes in the NLSE dynamics for Case III, and Fig. 4.9 depicts contour plots of −|ψε|as well as slice plots of |ψε(x, 0, t)| for showing sound wave propagation of the NLSEdynamics in Case IV. Based on Figs. 4.7-4.9 and additional computations not shownhere for brevity, we can draw the following conclusions:

(i). For Case I, there exits a critical time Tc depending on both d0 and ε suchthat when t < Tc the middle vortex (initially at the origin) will not move while theother two vortices rotate clockwise around the origin. This dynamics agrees very wellwith the NLSE dynamics in the whole plane [35, 36]. However, when t > Tc, themiddle vortex will begin to move towards one of the other two vortices and form asa vortex pair which will rotate with each other and also with the other single vortex


Fig. 4.8: Contour plots of |ψε(x, t)| with ε = 116 at different times for the NLSE

dynamics of a vortex lattice in Case III with different initial locations: d1 = d2 = 0.25(top two rows); d1 = 0.55, d2 = 0.25 (middle two rows); d1 = 0.25, d2 = 0.55 (bottomtwo rows) in section 4.4.


−5 0 50

0.56

1.12 t=0

x

|ψε (x

,0,t)|

−5 0 50

0.56

1.12 t=0.057

x

|ψε (x

,0,t)|

−5 0 50

0.56

1.12 t=0.174

x

|ψε (x

,0,t)|

−5 0 50

0.56

1.12 t=0.291

x

|ψε (x

,0,t)|

−5 0 50

0.54

1.08 t=0.372

x

|ψε (x

,0,t)|

−5 0 50

0.54

1.08 t=0.588

x

|ψε (x

,0,t)|

−5 0 50

0.54

1.08 t=0.777

x

!ψε (x

,0,t)|

−5 0 50

0.54

1.08 t=0.969

x

|ψε (x

,0,t)|

Fig. 4.9: Contour plots of −|ψε(x, t)| (left) and slice plots of |ψε(x, 0, t)| (right) atdifferent times for showing sound wave propagation under the NLSE dynamics of avortex lattice in Case IV with d0 = 0.5 and ε = 1

8 in section 4.4.

for a while. Then this pair will separate and one of them will form a new vortex pairwith the single vortex, leave the other one to be a new single vortex rotating withthem. This process will be repeated tautologically like three dancers exchange theirpartners alternatively. This shows that the boundary effect seems essentially affectthe vortex dynamics and interaction (cf. Fig. 4.7).

(ii). For Case II, similar as Case I, there exists a critical time Tc depending on d0and ε such that when t < Tc the middle vortex (initially at the origin) will not movewhile the other two vortices rotate couter-clockwise around the origin. This dynamicsagrees very well with the NLSE dynamics in the whole plane [35,36]. However, whent > Tc, the middle vortex will begin to move towards one of the other two vorticesand form as a vortex dipole which will move nearly parallel towards the boundary andmerge near the boundary. Sound waves will be created and reflected back into thedomain which drive the leftover vortex in the domain to move (cf. Fig. 4.7). Fromsection 4.1, we know that a single vortex in the NLSE with h(x) = 0 does not move,hence this example illustrates clearly the sound-vortex interaction. This also indicatesthat the reduced dynamical law fails completely after annihilation when t > Tc.

(iii). For Case III, the four vortices form as two vortex dipoles when t is small.Then the two dipoles will move outwards in opposite direction and finally the twovortices in each vortex dipole merge and annihilate at some place near the boundary.


If d1 = d2, the two vortex dipoles move symmetrically with respect to the line y = x,and merge some place near the top-right and bottom-left corners, respectively; ifd1 > d2, both of them will move towards the top-bottom boundary and merge nearthere; and if d1 < d2, both of them will move towards the side boundary and mergenear there. New waves are created after merging and they are reflected back into thedomain when they hit the boundary (cf. Fig. 4.8).

(iv). For Case IV, the vortex initially at the origin does not move due to sym-metry, while the other eight vortices rotate clockwise and move along two circle-liketrajectories (cf. Fig. 4.9). During the dynamics, sound waves are generated andthey propagate outwards and are reflected back into the domain when they hit theboundary. The distances between other vortices and the one centered at the originincrease when sound waves are radiated outwards; on the other hand, they decreaseand become even smaller than their initial distances when sound waves are reflectedby the boundary and move back into the domain (cf. Fig. 4.9). This example clearlyshows sound waves and their impact on the dynamics of vortices.

4.5. Radiation and sound wave. Here we study numerically how the radiationand sound waves affect the dynamics of quantized vortices in the NLSE dynamicsunder Dirichlet BC. To this end, we consider two types of perturbation.

Type I: Perturbation on the initial data, i.e. we take the initial data (1.2) as

(4.2) ψε(x, 0) = ψδ,ε0 (x) = ψε

0(x) + δe−10((|x|−0.08)2+y2), x = (x, y) ∈ Ω,

where ψε0 is given in (4.1) with h(x) ≡ 0,M = 2, n1 = n2 = 1 and x0

1 = −x02 = (0.1, 0),

i.e. we perturb the initial data for studying the interaction of a vortex pair by aGaussian function with amplitude δ. Then we take δ = ε and let ε go to 0, andsolve the NLSE (1.1) with initial condition (4.2) for the vortex centers x

δ,ε1 (t) and

xδ,ε2 (t) and compare them with those from the reduced dynamical law. We denote

dδ,εj (t) = |xδ,εj (t)− xrj(t)| for j = 1, 2 as the error. Fig. 4.10 depicts time evolution of

dδ,ε1 (t) for the case when δ = ε, i.e. small perturbation, and the case when δ = 0, i.e. noperturbation. From this figure, we can see that the dynamics of the two vortex centersunder the NLSE dynamics converge to those obtained from the reduced dynamicallaw when ε → 0 without perturbation (cf. Fig. 4.10 left). On the contrary, the twovortex centers under the NLSE dynamics do not converge to those obtained from thereduced dynamical law when ε → 0 with small perturbation (cf. Fig. 4.10 right).This clearly demonstrates radiation and sound wave effect on vortices in the NLSEdynamics with Dirichlet BC.

0 0.05 0.10

0.03

0.06

t

d 1δ,ε

δ=0, ε=1/64

δ=0, ε=1/80

δ=0, ε=1/100

0 0.05 0.10

0.008

0.016

t

d 1δ,ε

δ=ε=1/64

δ=ε=1/80

δ=ε=1/100

Fig. 4.10: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and perturbedinitial data (right) in section 4.5


−5 0 50

0.54

1.08t=0

x

|ψε (

x,0

,t)|

−5 0 50

0.54

1.08t=0.22

x

|ψε (

x,0

,t)|

−5 0 50

0.54

1.08t=0.52

x

|ψε (

x,0

,t)|

−5 0 50

0.54

1.08t=0.6

x

|ψε (

x,0

,t)|

−5 0 50

0.54

1.08 t=1.1

x

|ψε (

x,0

,t)|

−5 0 50

0.59

1.18 t=1.56

x

|ψε (

x,0

,t)|

−5 0 50

0.59

1.18 t=2.4

x

|ψε (

x,0

,t)|

−5 0 50

0.63

1.26 t=2.97

x

|ψε (

x,0

,t)|

Fig. 4.11: Surface plots of −|ψε(x, t)| (left) and slice plots of |ψε(x, 0, t)| (right) atdifferent times for showing sound wave propagation under the NLSE dynamics in adisk with ε = 1

4 and a perturbation in the potential in section 4.5.

Type II: Perturbation by an external potential, i.e. we replace V (x) ≡ 1 in (1.1)by V (x, t) = 1−W (x, t) with

(4.3) W (x, t) =

− sin(2t)2, t ∈ [0, 0.5],0, t > 0.5,

x ∈ Ω.

The initial data is chosen as (4.1) with M = 1, n1 = 1, x01 = (0, 0), Ω = B5(0)

and ε = 14 . In fact, the perturbation is introduced when t ∈ [0, 0.5] and is removed

after t = 0.5. Fig. 4.11 illustrates surface plots of −|ψε| and slice plots of ψε(x, 0, t)at different times for showing sound wave propagation. From Fig. 4.11, we cansee that the perturbed vortex configuration rotates and radiates sound waves. Thisagrees well with some former prediction in the whole plane, for example, in Lange andSchroers [19] for the case M = 2. The waves will be reflected back into the domainwhen they hit the boundary and then be absorbed by the vortex core. Then thevortex core will radiate new waves and the process is repeated tautologically. Thisprocess explicitly illustrates the radiation in the NLSE dynamics.


Remark 4.1. Based on this example and other numerical results not show herefor brevity, we can conclude that the vortex with winding number m = ±1 is dynam-ically stable under the NLSE dynamics in a bounded domain with a perturbation inthe initial data and/or external potential. Meanwhile, we also found numerically thatthe vortex with winding number m = 2 and ε = 1

32 is also dynamically stable undera perturbation in the external potential. Actually, Mirionescu [24] indicated that fora vortex with winding number |m| > 1, there exists a critical value εcm such that ifε < εcm , the vortex is unstable, otherwise the vortex is stable. It was also numericallyobserved that a vortex with |m| > 1 is unstable under a perturbation in the potentialbut stable under a perturbation in the initial data in the whole plane case [35]. Hence,it would be an interesting problem to investigate numerically how the stability of avortex depends on its winding number, value of ε and strength and/or type of theperturbation under the NLSE dynamics in bounded domains.

5. Numerical results under Neumann BC. In this section, we report numer-ical results for vortex interactions of the NLSE (1.1) under the homogeneous NeumannBC (1.4) and compare them with those obtained from the reduced dynamical laws(2.19) with (2.3). The initial condition ψε

0 in (1.2) is chosen as

(5.1) ψε0(x) = ψε

0(x, y) = eihn(x)M∏

j=1

φεnj(x− x0

j ), x = (x, y) ∈ Ω,

where hn(x) is harmonic function satisfying the Neumann BC

∂

∂nhn(x) = −

∂

∂n

M∑

l=1

nlθ(x− x0l ), x ∈ ∂Ω.

The NLSE (1.1) with (1.4), (1.2) and (5.1) is solved by the method TSCP presentedin section 3.

5.1. Single vortex. Here we present numerical results of the motion of a sin-gle quantized vortex under the NLSE (1.1) dynamics and its corresponding reduceddynamical laws, i.e. we take M = 1 and n1 = 1 in (5.1). Fig. 5.1 depicts trajectoryof the vortex center for different x0

1 in (5.1) when ε = 132 in (1.1) and dε1 for different

ε. From Fig. 5.1 and additional numerical results not shown here for brevity, we cansee that:

−1 0 1−1

0

1

x

y

0 1.8 3.60

0.02

0.04

t

d 1ε

ε=1/16

ε=1/25

ε=1/32

−1 0 1−1

0

1

x

y

0 1.8 3.60

0.02

0.04

t

d 1ε

ε=1/8

ε=1/16

ε=1/25

Fig. 5.1: Trajectory of the vortex center when ε = 132 and time evolution of dε1 for

different ε for the motion of a single vortex in NLSE under homogeneous NeumannBC with x0

1 = (0.35, 0.4) (left two) or x01 = (0, 0.2) (right two) in (5.1) in section 5.1.

(i). If x01 = (0, 0), the vortex will not move all the time, otherwise, the vortex

will move and its initial location x01 does not affect its motion qualitatively. Actually,


it moves periodically in a circle-like trajectory centered at the origin. This is quitedifferent from the situation in bounded domain with Dirichlet BC where the motionof a single vortex depends significantly on its initial location for some h(x). It is alsoquite different from the situation in the whole space where a single vortex doesn’tmove at all under the initial condition (5.1) when Ω = R

2.(ii). As ε → 0, the dynamics of the vortex center under the NLSE dynamics

converges uniformly in time to that of the reduced dynamical laws. In fact, basedon our extensive numerical experiments, the motion of the vortex center from thereduced dynamical laws agrees with that from the NLSE dynamics qualitatively when0 < ε < 1 and quantitatively when 0 < ε≪ 1.

5.2. Vortex pair. Here we present numerical results of the interaction of vortexpair under the NLSE (1.1) dynamics and its corresponding reduced dynamical laws,i.e. we take M = 2, n1 = n2 = 1 and x0

2 = −x01 = (d0, 0) with 0 < d0 < 1 in (5.1).

Fig. 5.2 depicts the trajectory of the vortex pair, time evolution of Eε(t), Eεkin(t),

xε1(t), x

ε2(t) and d

ε1(t) when ε =

132 in (1.1) and d0 = 0.5 in (5.1).

−1 0 1−1

0

1

x

y

0 0.92 1.8421

23

25

t

E ε

k i n

E ε

0 0.92 1.84−0.5

0.5

t

x 1ε or x

2ε

0 0.92 1.84−0.5

0.5

t

y 1ε or y

2ε

0 0.93 1.860

0.23

0.46

t

d 1ε

ε=1/8

ε=1/16

ε=1/32

Fig. 5.2: Trajectory of the vortex pair (left), time evolution of Eε and Eεkin (second),

xε1(t) and xε

2(t) (third), and dε1(t) (right) in the NLSE dynamics under homogeneous

Neumann BC with ε = 132 and d0 = 0.5 in section 5.2.

From Fig. 5.2 and additional numerical results not shown here for brevity, wecan draw the following conclusions for the interaction of vortex pair under the NLSEdynamics (1.1) with homogeneous Neumann BC:

(i) The total energy is conserved numerically very well during the dynamics.

(ii) The two vortices move periodically along a circle-like trajectory for all 0 <d0 < 1 and their trajectories are symmetric.

(iii) When ε→ 0, the dynamics of the two vortex centers under the NLSE dynam-ics converges uniformly in time to that of the reduced dynamical laws which verifiesnumerically the validation of the reduced dynamical laws in this case. In fact, basedon our extensive numerical experiments, the motion of the two vortex centers fromthe reduced dynamical laws agree with those from the NLSE dynamics qualitativelywhen 0 < ε < 1 and quantitatively when 0 < ε≪ 1.

5.3. Vortex dipole. Here we present numerical results of the interaction of vor-tex dipole under the NLSE (1.1) dynamics and its corresponding reduced dynamicallaws, i.e. we take M = 2, n1 = −n2 = −1, x0

2 = −x01 = (d0, 0) with different d0 and

ε = 132 in (1.1). Fig. 5.3 depicts the trajectory of the vortex dipole, time evolution of

xε1(t), x

ε2(t) and d

ε1(t).

From Fig. 5.3 and additional numerical results not shown here for brevity, we candraw the following conclusions for the interaction of vortex dipole under the NLSEdynamics (1.1) with homogeneous Neumann BC:


−1 0 1−1

0

1

x

y

0 1 2−1

1

t

x 1ε or

x 2ε

0 1 2−1

1

ty 1ε o

r y 2ε

−1 0 1−1

0

1

x

y

0 1 2−1

1

t

x 1ε or

x 2ε

0 1 2−1

1

t

y 1ε or

y 2ε

−1 0 1−1

0

1

x

y

0 0.65 1.3−1

1

t

x 1ε or

x 2ε

0 0.65 1.3−1

1

t

y 1ε or

y 2ε

0 1 20

0.09

0.18

t

d 1ε

0 1 20

0.07

0.14

t

d 1ε

ε=1/10

ε=1/20

ε=1/40

ε=1/16

ε=1/25

ε=1/32

Fig. 5.3: Trajectory and time evolution of xε1(t) and xε

2(t) for d0 = 0.25 (top left two),d0 = 0.7 (top right two) and d0 = 0.1 (bottom left two) and time evolution of dε1(t)for d0 = 0.25 and d0 = 0.7 (bottom right two) in section 5.3.

(i) The total energy is conserved numerically very well during the dynamics.

(ii) The pattern of the motion of the two vortices depends on their initial locations.

(iii) The two vortices will move symmetrically (and periodically if they are wellseparated) with respect to y-axis. Moreover, there exists a critical value drc = dεc = dcfor 0 < ε < 1, which is found numerically as dc = 0.5, such that if initially d0 < dc,the two vortices will move firstly upwards to the top boundary, then turn outwardsto the side boundary and finally move counter-clockwise and clockwise, respectively(cf. Fig. 5.3). While if d0 > dc, then they will move firstly downwards to the bottomboundary, then turn inwards to the domain and finally move counter-clockwise andclockwise, respectively (cf. Fig. 5.3). Certainly, when d0 = 0.5, the vortex dipoledoes not move due to symmetry.

(iv) For fixed 0 < ε < 1, there exists another critical value dεc satisfying limε→0 dεc =

0, such that if d0 < dεc, the vortex dipole in the NLSE dynamics will merge at a finitetime Tc depending on ε and d0 (cf. Fig. 5.3). However, the vortex dipole from thereduced dynamical laws never merges at finite time. Hence, the reduced dynamicallaws fail qualitatively if 0 < d0 < dε0.

(v). For fixed d0, when ε→ 0, the dynamics of the two vortex centers in the NLSEdynamics converges uniformly in time to that of the reduced dynamical laws beforethey merge (cf. Figs. 5.3) which verifies numerically the validation of the reduceddynamical laws in this case. In fact, based on our extensive numerical experiments,the motion of the two vortex centers from the reduced dynamical laws agree withthose from the NLSE dynamics qualitatively when 0 < ε < 1 and quantitatively when0 < ε≪ 1.

5.4. Vortex lattice. Here we present numerical studies on the dynamics ofvortex lattices in the NLSE (1.1) with homogeneous Neumann BC, i.e. we choose theinitial data (1.2) as (5.1) and study four cases:

Case I. M = 3, n1 = n2 = n3 = 1, x01 = −x0

3 = (d0, 0) and x02 = (0, 0).

Case II. M = 4, n1 = n2 = n3 = n4 = 1, x01 = −x0

2 = (d1, 0) and x03 = −x0

4 =(d2, 0) with 0 < d1 6= d2 < 1.


−1 0 1−1

0

1

x

y

−1 0 1−1

0

1

x

y

−0.4 0 0.4−0.4

0

0.4

x

y

0 0.13 0.27 0.4−0.3

0

0.3

0.6

t

x

3ε(t)

y3ε(t)

x2ε(t)

y2ε(t)

0.4 0.43 0.46 0.49

−0.3

0

0.3

0.6

t

x

3ε(t)

y3ε(t)

x2ε(t)

y2ε(t)

0.49 0.81 1.13 1.45

−0.3

0

0.3

0.6

t

x

3ε(t)

y3ε(t)

x2ε(t)

yε(t)

Fig. 5.4: Trajectory of the vortex xε1 (blue line), xε

2 (dark dash-dot line) and xε3

(red dash line) and their corresponding time evolution for Case I during small time(left column), intermediate time (middle column) and large time (right column) withε = 1

40 and d0 = 0.25 in section 5.4.

Case III. M = 4, n1 = n2 = −n3 = −n4 = 1, x01 = −x0

2 = (d1, 0) and x03 =

−x04 = (0, d2) with 0 < d1, d2 < 1.Case IV. M = 9, n1 = · · · = n9 = 1, and the vortex centers are initially located

on a 3× 3 uniform mesh points for the rectangle [−d0, d0]× [−d0, d0] with 0 < d0 < 1.Fig. 5.4 shows trajectory and time evolution of xε

1(t), xε2(t) and xε

3(t) in the NLSEdynamics for Case I. Fig. 5.5 depicts contour plots of |ψε| at different times in NLSEdynamics for Cases II and III, and Fig. 5.6 shows contour plots of −|ψε| and sliceplots of |ψ(0, y, t)| in NLSE dynamics for Case IV to show sound wave propagation.Based on Figs. 5.4-5.6 and additional results not shown here for brevity, we can drawthe following conclusions:

(i). For Case I, there exits a critical time Tc depending on both d0 and ε suchthat when t < Tc the middle vortex (initially at the origin) will not move while theother two vortices rotate clockwise around the origin. This dynamics agrees very wellwith the NLSE dynamics in the whole plane [35, 36]. However, when t > Tc, themiddle vortex will begin to move towards one of the other two vortices and form asa vortex pair which will rotate with each other and also with the other single vortexfor a while. Then the two vortices in the pair will separate, one of them will form anew vortex pair with the third single vortex leaving the other one to be a new singlevortex that rotates around them. This process will repeat tautologically like threedancers exchange their partners alternatively. This indicates that the boundary effectessentially affects the vortex dynamics and interaction (cf. Fig. 5.4).

(ii). For Case II, the four vortices form as two vortex pairs when t is small. Thesetwo pairs rotate with each other clockwise, meanwhile, the two vortices in each pairalso rotate with each other clockwise, and radiations and sound waves are emitted.The sound waves propagate radially and are reflected back into the domain when theyhit the boundary, which push the two vortex pairs get closer. When the two vortexpairs get close enough, the two vortices with smallest distance among the four form anew vortex pair and leave the rest two as single vortex individually. The vortex pair


Fig. 5.5: Contour plots of |ψε(x, t)| with ε = 116 at different times for the NLSE

dynamics of a vortex lattice for Case II with d1 = 0.6, d2 = 0.3 (top two rows) andCase III with d1 = d2 = 0.3 (bottom two rows) in section 5.4.

rotates around the origin. This process is iteratively repeated during the dynamics(cf. Fig. 5.5 top two rows).

(iii). For Case III, when t is small, the four vortices form as two vortex dipolesand they move symmetrically with respect to the line y = −x towards the top rightand bottom left corners, respectively. Meanwhile, the two vortices in each dipolemove symmetrically with respect to the line y = x. After a while and when the twodipoles arrive at some places near the corners, the two vortices in each dipole splitwith each other and re-formulate two different dipoles. After this, the two vorticesin each dipole move symmetrically with respect to the line y = −x, and the two newdipoles then move symmetrically with respect to the line y = x towards their initiallocations. This process is then repeated periodically (cf. Fig. 5.5 bottom two rows).

(iv). For Case IV, the vortex initially centered at the origin does not move dueto symmetry, and the other eight vortices rotate clockwise and move along two circle-


−1 0 10

0.53

1.06t=0

y

|ψε (0

,y,t)

|

−1 0 10

0.56

1.12 t=0.006

y

|ψε (0

,y,t)

|

−1 0 10

0.58

1.16t=0.014

y

|ψε (0

,y,t)

|

−1 0 10

0.58

1.16t=0.036

y

|ψε (0

,y,t)

|

−1 0 10

0.55

1.1t=0.046

y

|ψε (0

,y,t)

|

−1 0 10

0.54

1.08t=0.051

y

|ψε (0

,y,t)

−1 0 10

0.59

1.18 t=0.07

y

|ψε (0

,y,t)

−1 0 10

0.59

1.18t=0.264

y

|ψε (0

,y,t)

Fig. 5.6: Contour plots of −|ψε(x, t)| (left) and slice plots of |ψε(0, y, t)| (right) atdifferent times under the NLSE dynamics of a vortex lattice in Case IV with d0 = 0.15and ε = 1

40 for showing sound wave propagation in section 5.4.

like trajectories (cf. Fig. 5.6). During the dynamics, sound waves are generated andthey propagate outwards. Some of the sound waves will exit out of the domain whileothers are reflected back into the domain when they hit the boundary. The distancesbetween the one located at the origin and the other vortices become larger when thesound waves are radiated outwards, while they decrease when the sound waves arereflected from the boundary and move back into the domain (cf. Fig. 5.6).

5.5. Radiation and sound wave. Here we study numerically how the radiationand sound waves affect the dynamics of quantized vortices in the NLSE dynamicsunder homogeneous Neumann BC. To this end, we take the initial data (1.2) as (4.2)with ψε

0 chosen as (5.1) with M = 2, n1 = n2 = 1 and x01 = −x0

2 = (0.1, 0), i.e. weperturb the initial data for studying the interaction of a vortex pair by a Gaussianfunction with amplitude δ. Then we take δ = ε and let ε goes to 0, and solve the NLSE(1.1) with initial condition (4.2) for the vortex centers xδ,ε

1 (t) and xδ,ε2 (t) and compare

them with those from the reduced dynamical law. We denote dδ,εj (t) = |xδ,εj (t)−xrj(t)|

for j = 1, 2 as the error. Fig. 5.7 depicts time evolution of dδ,ε1 (t) for the case whenδ = ε, i.e. small perturbation, and the case when δ = 0, i.e. no perturbation.From this figure, we can see that the dynamics of the two vortex centers under theNLSE dynamics converge to those obtained from the reduced dynamical law when


ε → 0 without perturbation (cf. Fig. 5.7 left). On the contrary, the two vortexcenters under the NLSE dynamics do not converge to those obtained from the reduceddynamical law when ε → 0 with small perturbation (cf. Fig. 5.7 right). This clearlydemonstrates radiation and sound wave effect on vortices under the NLSE dynamicswith homogenous Neumann BC.

0 0.035 0.070

0.02

0.04

t

d 1δ,ε

δ=0, ε=1/64

δ=0, ε=1/80

δ=0, ε=1/100

0 0.035 0.070

0.006

0.012

td 1δ,

ε

δ=ε=1/64

δ=ε=1/80

δ=ε=1/100

Fig. 5.7: Time evolution of dδ,ε1 (t) for non-perturbed initial data (left) and perturbedinitial data (right) in section 5.5

6. Conclusion. We studied numerically quantized vortex dynamics and theirinteraction of the nonlinear Schrodinger equation (NLSE) or the Gross-Pitaevskiiequation with a dimensionless parameter 0 < ε < 1 on bounded domains under eitherDirichlet or homogenous Neumann BC for superfluidity. Based on our extensivenumerical results, we have: (i) verified that the dynamics of vortex centers under theNLSE dynamics converges to that of the reduced dynamical laws when ε→ 0 beforethey collide and/or move too close to the boundary; (ii) identified parameter regimeswhere the reduced dynamical laws agree quantitatively and/or qualitatively as wellas fail to agree with those from the NLSE dynamics; (iii) observed clearly radiationand sound wave effect on quantized vortex interaction under the NLSE dynamics onbounded domains.

AcknowledgmentPart of this work was done when the first author was visiting Beijing Computa-

tional Science Research Center in 2012 and he also acknowledges Professor Fang-HuaLin for stimulating discussion.

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